Eulerian viewpoint

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

In an engineering view the ensemble of system points moving through phase space behaves much like a fluid in a multidimensional space, and there are numerous similarities between our imagination of the ensemble and the well known notions of fluid dynamics [35]. Then, the substantial derivative in fluid dynamics corresponds to a derivative of the density as we follow the motion of a particular differential volume of the ensemble in time. The material derivative is thus similar to the Lagrangian picture in fluid d3mamics in which individual particles are followed in time. The partial derivative is defined at fixed (q,p). It can be interpreted as if we consider a particular fixed control volume in phase space and measure the time variation of the density as the ensemble of system points flows by us. The partial derivative at a fixed point in phase space thus resembles the Eulerian viewpoint in fluid dynamics. 

Equations 7.76 and 7.78, which are alternative forms of the x-directed momentum balance, represent two ways of regarding fluid mechanics problems. In Eq. 7.76 the left-hand side is the time rate of change of momentum of the fluid contained in an infinitesimal volume at some fixed point in space. The right-hand terms, in order, are the increase of momentum due to flow of matter into and out of this volume and the net forces on the system due to gravity, normal, and shear forces. This viewpoint, which is the viewpoint of an observer fixed in space, is called the eulerian viewpoint. 

Although it is difficult to induce turbulence (so-called Eulerian chaos) in microchannels, the mixing performance obtained in low Reynolds number flow regimes can be enhanced via the chaotic advection mechanism (or so-called Lagrangian chaos). Chaotic advection occurs in regular, smooth (from a Eulerian viewpoint)... 

There are two ways to look at a cloud of diffusing molecules the Eulerian viewpoint or the Lagrangian viewpoint. In the Eulerian viewpoint, the diffusion equations would be derived from a consideration of concentrations and flux at a fixed point in space. Such quantities are easily observable. It is possible to arrive at the same results fi om the Lagrangian viewpoint, which focuses on the history of the random movements of the diffusing material. Statistical properties of the random motions would have to be mathematically described to produce useful results. 

There are two viewpoints that one may take to describe the fluid motion the Lagrangian and the Eulerian viewpoints. 

In the Eulerian viewpoint, the quantities in the flow field are described as a function of fixed position x,- and time t. Specifically, the flow velocity is described... 

For steady-state flows (here we consider the steady-state in the Eulerian viewpoint. In a steady Eulerian velocity field (which is not necessarily Lagrangian steady), 0M,/0f = 0. For the analysis of polymer melt flow occurring at a very low Reynolds number, the inertia term puj dujldxj will usually be neglected. The body force effect is usually negligible too. This reduces the equations of motion to... 

Acoustic streaming is a secondary steady flow generated from the primary oscillatory flow. It includes not only the Eulerian streaming flow caused by the fluid dynamical interaction but also the Stokes drift flow which arises from a purely kinematic viewpoint. 

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